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Actions of nonamenable groups on Z-stable C*-algebras

Gardella, Eusebio and Lupini, Martino (2018) Actions of nonamenable groups on Z-stable C*-algebras. . (Submitted)

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We study strongly outer actions of discrete groups on C*-algebras in relation to (non)amenability. In contrast to related results for amenable groups, where uniqueness of strongly outer actions on the Jiang-Su algebra is expected, we show that uniqueness fails for all nonamenable groups, and that the failure is drastic. Our main result implies that if G contains a copy of the free group, then there exist uncountable many, non-cocycle conjugate strongly outer actions of G on any Jiang-Su stable tracial C*-algebra. Similar conclusions apply for outer actions on McDuff tracial von Neumann algebras. We moreover show that G is amenable if and only if the Bernoulli shift on a finite strongly self-absorbing C*-algebra absorbs the trivial action on the Jiang-Su algebra. Our methods consist in a careful study of weak containments of the Koopman representations of different Bernoulli-type actions.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Lupini, Martino0000-0003-1588-7057
Alternate Title:Group amenability and actions on Z-stable C*-algebras
Additional Information:This work was initiated during a visit of the first named author to the second at the California Institute of Technology in January 2017, and was continued during a visit of both authors to the Centre de Recerca Matemàtica in March 2017 in occasion of the Intensive Research Programme on Operator Algebras. The authors gratefully acknowledge the hospitality and the financial support of both institutions. The first named author was partially funded by SFB 878 Groups, Geometry and Actions, and by a postdoctoral fellowship from the Humboldt Foundation, and the second named author was partially supported by the NSF Grant DMS-1600186.
Funding AgencyGrant Number
Centre de Recerca MatemàticaUNSPECIFIED
Deutsche Forschungsgemeinschaft (DFG)SFB 878
Alexander von Humboldt FoundationUNSPECIFIED
Subject Keywords:Amenable group, free group, Bernoulli shift, GNS construction, weak containment, cocycle equivalence, Jiang-Su algebra, hyperfinite II1-factor.
Classification Code:2000 Mathematics Subject Classification. Primary 46L55, 54H05; Secondary 03E15, 37A55.
Record Number:CaltechAUTHORS:20180410-091307748
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:85719
Deposited By: Tony Diaz
Deposited On:10 Apr 2018 18:22
Last Modified:03 Oct 2019 19:34

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